User blog:Holomanga/Number System (from V
A number system is a bunch of things, any two of which can be compared with the relation < to give a value of true or false. Zero ∅, the empty set, is the smallest number, called 0. Finite S(n)=n+1 is defined as n∪{n}. This means that each number is the collection of all smaller numbers. This applies generally. nn/m. * Reals ℝ: Cauchy sequences of rationals * Complex numbers ℂ: pairs of reals with operations given by the Cayley-Dickinson construction Infinite ℕ is the union of the numbers that can be constructed by composing ∅ with S(n) a finite number of times. These are called finite numbers. The set N is also a number, called ω. Multiples of ω, notated by ω.n, are the unions of the numbers that can be constructed by composing ω.(n-1) with S(n) a finite number of times. Other operations on ω are constructed similarly. Powers of ω, notated by ωn, are the unions of all ωn-1.m for finite m. ωω is the union of all ωm for finite m. Repeated powers are constructed similarly; ωωn is the union of all ωωn-1.m for finite m. This continues generally. ε0 is the union of all ω↑↑ω. Countable ordinals are well-studied and many of them have names. ω1 is the union of all numbers with cardinality |m|<ℵ1, and generally ωn is the union of all numbers with cardinality |m|<ℵn. Unsettling Ord is the union of all sets that are numbers. The number is called Ω. This is a proper class. Further numbers are defined similarly to the infinite numbers, but with < being the element-of relation over higher-order collections such as classes and conglomerates. Squandered The union of all collections is the squandered number ξ, pronounced “Cantor’s Absolute Infinity”. As the union of all collections, ξ contains itself, so ξ<ξ. This is a fixed point of the power “set” operation, since the collection of subcollections of the union of all collections is the union of all collections: ℘(ξ)=ξ. Larger squandered numbers are defined such that ξm<ξn, but ξn≮ξm, if m≤n, with the squandered number ξ=ξ0. That ξ is defined such that there exist no collections satisfying this property is a contradiction that by no means perturbs the squandered numbers. These pseudocollections are not well-founded because they contain themselves. Voluntary One can form circles in the ordering relation between numbers. ξn<ξn is an order-one circle, because it only has one element. Higher circles can be created, such as some ξn<ξm<ξn (with n≠m), giving an order-two circle due to containing two elements (note that n and m would also make an order-two circle, which means that these are fixed points of the squandered indexing function). The smallest order-n circle is notated by ��n. ��0=0, ��1=ξ. Such numbers with 2≤n are called voluntary numbers. The limit of voluntary numbers is ∞ALL (the infinity over ALL), defined such that n≤∞ALL for any n, meaning that all greater numbers form an arbitrarily large circle with ∞ALL. Category:Blog posts